nLab
solenoid

Solenoids

Idea

A simple example of a solenoid is the dyadic solenoid. This is the inverse limit of the inverse sequence, (X(n),pn)(X(n),p_n), in which each X(n)X(n) is a copy of the circleS1S^1 and each ‘structure map’ pn:X(n)→X(n−1)p_n\colon X(n) \to X(n-1) is given by the squaring map on the circle, that is eiθ↦e2iθe^{i\theta}\mapsto e^{2i\theta} (viewing S1S^1 as the unit circle in the complex plane).

We will restrict attention, for the moment, to the class of solenoids defined by inverse sequences of circles:

General definition

Let P=(r1,…,rn,…)P = (r_1,\ldots, r_n,\ldots) be a sequence of prime numbers. The PP-adic-solenoid is the space,

SP=Lim(X(n),pn,ℕ)S_P = Lim (X(n),p_n,\mathbb{N})

where for each nn, X(n)=S1X(n)= S^1 and (considering S1S^1 as the unit circle in ℂ\mathbb{C}), pn(z)=zrnp_n(z) = z^{r_n}.

Solenoids and shape

In shape theory, the solenoids provide good examples of non-stable spaces. If one takes the Čech fundamental group? of SPS_P, then it is the limit of the inverse sequence of the fundamental groups of the X(n)X(n) together with the induced homomorphisms. It thus is trivial as it contains all integers that are divisible by all the primes in the sequence, and that an infinite number of times! The Strong Shape fundamental group of SPS_P is not trivial as it involves the first derived functor Lim(1)Lim^{(1)} of the inverse sequence of groups as well.

References

For solenoids in many settings, see the Wikipedia page, which also has some very neat graphics!